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Calculus

There are two components to calculus. One is the measure the rate of change at any given point on a curve. This rate of change is called the derivative. The simplest example of a rate of change of a function is the slope of a line. We take this one step further to get the rate of change at a point on a line. The other part of calculus is used to measure the exact area under a curve. This is called the integral. If you wanted to find the area of a semicircle, you could use integration to get the answer.

The two parts; the derivative and the integral are inverse functions of each other. That is, they cancel each other out.
Just as (x2)1/2=x,
the derivative of (integral (x)) = x and
derivative of (integral (f (x)) = f(x).

The derivative is a composite function. This means it is a function acting on another funcion. In fact, the function, is the input instead of just x. The derivative, then takes a type of formula and turns it into another simiilar type of formula. So, a polynomial will always yield a polynomial derivative. A trigonomic function will always yield a trigonomic derivative. There are a few exceptions, but this is generally the case. This is also true for the integral

 

Geometrically, the derivative can be perceived as the slope of the tangent line to a curve at a given point. This is roughly how steep the curve is at a given point. We can easily find the rate of change of a line just by finding the slope. But, most formulas are not as simple as a line and they're usually curved. We use the basic formula of a line to get the derivative. If you remember the slope of a line is:

y2-y1

x2-x1

where the change in height is divided by the change in width. The derivative is derived by using this formula and taking x2to be infinitely close to x1. When plugging differnt formulas into this formula, they are translated to another similar type formula. The general formula for the derivative is:
The derivative of f(x) = limit as x2 goes to x1 of the formula:

f(x2)-f(x)1


x2-x1

This can be written in shorthand as:

f'(x) = lim f(x2) - f(x1)

x2 -->x1 x2 - x1

So, what would happen if we plug in a formula like x2 in the above equation? Remember, we're dealing with a composite funcion, so f(x) is substituted instead of x. So, our derivative of f(x) = x2 is:



f'(x) = lim x22 - x12

x2 --> x1 x2 - x1


= 2x





There is a proof to this, but I don't think that is necessary in this brief introduction to calculus. But, did you notice what happened? (x<SUP2< sup>)' becomes 2x. The more general formula for the derivative of a polynomial term ,with no constant term in front of the x, is
(xn)' = nxn-1So, if f(x) = x4, then f'(x) = (4)x3 = 4x3. And, for any polynomial term with a constant factor as in f(x) = 5x3, the formula is
f'(x) = (3)(5)x2 = 15x2 where you take n times the constantt. So, the general formula for the derivative for the polynomial is
(cxn)'=(c)(n)xn-1
where c is any constant.There are other types of derivatives. Some of the main ones are:

If f(x) = ex, then f'(x) = ex
If f(x) = sin x, then f'(x) = cos xand if f(x) = sin nx, then f'(x) = n cos nx
If f(x) = cos x, then f'(x) = -sin x andif f(x) = cos nx, then f'(x) = -nsin xThere are many other types, but these are a few of the main ones to get you started.